Method Of Designing Passive Rc Complex Filter Of Hartley Radio Receiver

ABSTRACT

An object of this invention is to provide a method of designing a complex transfer function which can be realized in a passive RC complex filter at the same time while perfectly succeeding to features of a prototype lowpass characteristic. In this invention, as a first step, a prototype lowpass characteristic F(p) having a pole on a unit circle is designed. Next, as a second step, a bilinear variable transformation expressed by p=j(s−j)/(s+j) is performed with respect to the prototype characteristic F(p) to derive a complex coefficient transfer function G(s). As a third step, a passive RC complex filter H(s) is designed based upon the complex coefficient transfer function G(s).

TECHNICAL FIELD

This invention relates to a method of designing a passive RC complexfilter employed in a Hartley Image-rejection radio receiver for wideningan Image-rejection frequency range.

BACKGROUND OF THE INVENTION

Although heterodyne receivers typically constitute a receiverarchitecture in currently available radio appliances, those heterodynereceivers have an essential problem of image interference. Imageinterference is a phenomenon in which two RF signals having asymmetrical relationship between a high frequency side and a lowfrequency side with respect to a local oscillator frequency “ω0” areconverted into the same intermediate frequency (IF) range by a downconversion.

In FIG. 2, for the sake of convenience, it is so assumed that a bandpassfilter 100 is not provided. An input signal “X” of a radio frequencyband (ω₀+Δω) is down-converted by a mixer 101 by utilizing the localoscillator frequency “ω₀”. A low frequency component among the mixeroutput signal is extracted by a lowpass filter 102, whereby anintermediate frequency signal “v_(1/)” with “Δω” as a carrier isobtained. On the other hand, if an interference wave “Y” is present onthe low frequency side (ω₀−Δω), the interference wave “Y” is mixed asrepresented in Equation 1 and thus cannot be discriminated. As a result,reception performance is deteriorated. $\begin{matrix}{{v_{RF} = {{X\quad{\cos\left( {{\omega_{0}t} + {{\Delta\omega}\quad t}} \right)}} + {Y\quad{\cos\left( {{\omega_{0}t} - {{\Delta\omega}\quad t}} \right)}}}}{v_{LI} = {\cos\quad\omega_{0}t}}{v_{1I} = \frac{{X\quad{\cos\left( {{\Delta\omega}\quad t} \right)}} + {Y\quad{\cos\left( {{\Delta\omega}\quad t} \right)}}}{2}}} & {{Equation}\quad 1}\end{matrix}$

Normally, in order to prevent the image interference, the bandpassfilter 100 which passes only the frequency band (ω₀+Δω) is required at afront stage of the mixer. However, since the filter can be hardlyintegrated and a passband thereof is fixed, it is considerably difficultto apply the bandpass filter to a plurality of radio systems havingdifferent bands.

Image-rejection receivers are effective for removing the bandpass filter100 from the receivers. As one of those effective receiver systems, aHartley receiver shown in FIG. 3 is known. A basic idea of rejecting animage is expressed in Equation 2. An input signal v_(RF) isquadrature-down-converted by mixers 101 and 104 by employing localoscillator signals v_(LI) and v_(LQ) which are orthogonal to each other.Next, low frequency components v_(1/) and v_(1Q) are extracted bylowpass filters 102 and 105. After one of those signals is shifted by 90degrees by a phase shifter 106, the phase-shifted signal is added andsynthesized with the other signal by adding means 107, whereby the imagecan be removed. Although a 0-degree phase shifter 103 is inserted inFIG. 3, the lowpass filter 102 may be directly coupled to the addingmeans 107 in an actual case. $\begin{matrix}{{v_{LQ} = {\sin\quad\omega_{0}t}}{v_{2I} = v_{1I}}{{v_{1\quad Q} = \frac{{{- X}\quad{\sin\left( {{\Delta\omega}\quad t} \right)}} + {Y\quad{\sin\left( {{\Delta\omega}\quad t} \right)}}}{2}},{v_{2Q} = \frac{{X\quad{\cos\left( {{\Delta\omega}\quad t} \right)}} - {Y\quad{\cos\left( {{\Delta\omega}\quad t} \right)}}}{2}}}{v_{IF} = {{v_{2I} + v_{2Q}} = {X\quad{\cos\left( {{\Delta\omega}\quad t} \right)}}}}} & {{Equation}\quad 2}\end{matrix}$

In an actual circuit, as shown in FIG. 4, a 45-degree phase shifter 106b is employed instead of the 90-degree phase shifter 106, and a minus45-degree phase shifter 103 b is employed instead of the 0-degree phaseshifter 103. These phase shifters 106 b and 103 b are substituted by afirst-order RC lowpass filter and a first-order RC highpass filter in anapproximated manner, in which the cut-off frequency is set to ω_(c)=Δω.Therefore, the only frequency at which the image can be completelyrejected is the frequency of ω=Δω in which the amplitude of the lowpassfilter is made coincident with the amplitude of the highpass filter.With respect to the above-mentioned image interference andImage-rejection receiver, a detailed description thereof is made in “RFMICROELECTRONICS” edited/translated by Tadahiro Kuroda, MARUZEN, 2002.

It should be noted that when differential mixers 101 b and 104 b areemployed, as shown in FIG. 5, because signals v_(1/), v_(1Q), −v_(1/),and −v_(1Q) of four phases can be obtained, the phases of which areshifted by 90 degrees, respectively, and a polyphase filter 108 can beemployed. Four resistance values and four capacitance values are equalto each other. In this case, the adding function is realized by a signalsuperimposing effect of the polyphase filter. The highpass filteringoperation is effected with respect to the signal v_(1/), and the lowpassfiltering operation is effected with respect to the signal v_(1Q), whichare equivalent to the location of FIG. 4. It should be noted that sincethe phases of the output signals are merely shifted by 90 degrees, anyof the four terminals may be selected.

The bands of the Hartley receivers have been described in a qualitativemanner. Quantitatively, as described in “Explicit Transfer Function ofRC Polyphase Filter for Wireless Transceiver Analog Front-End” by H.Kobayashi, J. Kang, T. Kitahara, S. Takigami, and H. Sakamura, 2002 IEEEAsia-Pacific Conference on ASICs, pp. 137-140, Taipei, Taiwan (August2002), it is only necessary that a complex transfer function “H(s)”, inwhich one of a highpass characteristic and a lowpass characteristic isset as a real part “H_(r)(s)” and the other is set as an imaginary part“H_(i)(s)”, be defined, and a frequency response be observed. At thistime, a negative frequency becomes a response with respect to an image,whereas a positive frequency becomes a response with respect to adesirable wave. The complex transfer function H(s) using a first-orderfilter is expressed in Equation 3, and a frequency response obtained bynormalizing “ω_(c)” is shown in FIG. 6. $\begin{matrix}{{{H_{r}(s)} = \frac{s}{s + \omega_{c}}},{{H_{i}(s)} = \frac{\omega_{c}}{s + \omega_{c}}},{{H(s)} = \frac{s + {j\omega}_{c}}{s + \omega_{c}}}} & {{Equation}\quad 3}\end{matrix}$

Expansion of bandwidth of the Hartley receivers result in designingproblems of passive RC complex filters, and higher-order complextransfer functions having wide band frequency responses in both apassband and a stopband must be designed. One of the conventionaltechniques for designing the higher-order complex transfer functions isdescribed in “Low-IF topologies for high-performance analog front endsof fully integrated receivers” by J. Crols and M. S. Steyaert, IEEETrans Circuits Syst.-II, vol. 45, pp. 269-282, March 1998.

In the conventional technique, first of all, a proper prototype lowpasscharacteristic is designed. As to the prototype lowpass characteristic,various sorts of characteristics are known, for instance, a Butterworthfilter, and various higher-order characteristics can be readilydesigned.

Next, a variable transformation is performed with respect to theprototype lowpass filter so as to shift the frequency response to theside of the positive frequency on the frequency axis. Because of theshift operation, a bandpass type complex transfer function with which apositive frequency band becomes a passband and a negative frequencybecomes a stopband is obtained. This transfer function succeeds to theshape of the prototype lowpass characteristic. However, with thismethod, since the transfer function has a complex pole, such arestriction that a passive RC complex filter can only have a negativereal pole in a simple root cannot be satisfied. As a result, there is nochoice but to realize the prototype lowpass characteristic by an activefilter, resulting in demerits in terms of noise and power consumption ofactive elements.

Another method is described in “RC Polyphase Filter with Flat GainCharacteristic” by Kazuyuki Wada and Yoshiaki Tadokoro, Proceedings ofthe 2003 IEEE International Symposium on Circuits and Systems, Vol. I,pp. 537-540, May 2003. In this conventional technique, because theelement values are directly designed based on the assumption of thestructure of the higher-order RC polyphase filter, an active element isnot required. However, although the frequency of the transfer zeropoint, namely, the resistance/capacitance products at the respectivestages are clearly given based upon the equi-ripple model, theresistance values are, properly determined based upon the arbitraryconstant “α.” In other words, although the numerator of the transferfunction is perfectly designed, the denominator thereof is imperfect.Accordingly, the flatness of the passband cannot be completelyguaranteed.

SUMMARY OF THE INVENTION

This invention has been made to overcome drawbacks in the conventionaldesigning methods. That is, an object of this invention is to provide amethod of designing a complex transfer function which can be realized ina passive RC complex filter at the same time while perfectly succeedingto the feature of the prototype lowpass characteristic.

In this invention, as a first step, a prototype lowpass transferfunction F(p) having a pole on a unit circle is designed. For the F(p),a Butterworth filter (Butterworth characteristic) or a elliptic filter(simultaneous Chebyshev characteristic) in which a maximum passband loss“α” and a minimum stopband loss “A” satisfy a condition equation ofα=A/√(A²−1) is employed.

Next, as a second step, a bilinear variable transformation expressed byp=j(s−j)/(s+j) is performed with respect to the prototype transferfunction F(p) so as to derive a complex coefficient transfer functionG(s). The complex coefficient transfer function G(s) has a pole in asimple root on a negative real axis, and a passband is present in apositive frequency and a stopband is present in a negative frequency.

As a third step, a passive RC complex filter H(s) having a transferfunction identical to the complex coefficient transfer function G(s) isdesigned, and thereafter, an impedance scaling is performed, whereby thedesigning method is accomplished. The above-mentioned designing flow isshown in FIG. 1.

According to the above-mentioned designing method, while perfectlytaking the feature of the prototype lowpass transfer function F(p), ahigher-order complex transfer function G(s) which can be realized by apassive RC complex filter can be simultaneously designed. A Hartleyreceiver using the higher-order complex transfer function G(s) canreject images over a broadband and can maintain flatness with respect toa desirable wave at the same time. In addition, the Hartley receiver isexcellent in terms of circuit noise and power consumption, as comparedwith those using an active filter.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention can be appreciated by the description whichfollows in conjunction with the following figures, wherein:

FIG. 1 is a designing flow chart of this invention.

FIG. 2 is a structural diagram of a heterodyne receiver.

FIG. 3 is a structural diagram of a Hartley receiver.

FIG. 4 is a structural diagram of another embodiment of a Hartleyreceiver.

FIG. 5 shows a polyphase structure of the Hartley receiver.

FIG. 6 shows a frequency response of a first-order complex filter.

FIG. 7 shows a pole location of a Butterworth filter.

FIG. 8 shows a frequency response of the Butterworth filter.

FIG. 9 shows a relationship to be satisfied by a maximum passband loss“α” and a minimum stopband loss “A”.

FIG. 10 shows a pole location of an elliptic filter (A=30 dB).

FIG. 11 shows a pole location of the elliptic filter (A=40 dB).

FIG. 12 shows a pole location of the elliptic filter (A=50 dB).

FIG. 13 shows a frequency response of the elliptic filter (A=30 dB).

FIG. 14 shows a frequency response of the elliptic filter (A=40 dB).

FIG. 15 shows a frequency response of the elliptic filter (A=50 dB).

FIG. 16 shows a frequency response of a complex transfer function basedupon the Butterworth filter.

FIG. 17 shows a frequency response of the complex transfer functionbased upon the elliptic filter (A=30 dB).

FIG. 18 shows a frequency response of the complex transfer functionbased upon the elliptic filter (A=40 dB).

FIG. 19 shows a frequency response of the complex transfer functionbased upon the elliptic filter (A=50 dB).

FIG. 20 shows a fifth-order RC polyphase filter.

FIG. 21 shows Another Embodiment 1 of a third-order complex filter(elliptic filter of A=30 dB).

FIG. 22 shows a frequency response of Another Embodiment 1 of thethird-order complex filter.

FIG. 23 shows Another Embodiment 2 of the third-order complex filter(elliptic filter of A=30 dB).

FIG. 24 shows a frequency response of Another Embodiment 2 of thethird-order complex filter.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

First, a description is made of a prototype lowpass transfer function“F(p)” having a pole on a unit circumference. A first characteristiccorresponds to a Butterworth characteristic. A transfer function of ann-th order Butterworth filter in which a power half value (3-dB loss)frequency is normalized is expressed by the following equation:$\begin{matrix}{{F(p)} = \left\{ {\begin{matrix}{{\frac{1}{p + 1}{\prod\limits_{k = 1}^{{({n - 1})}/2}\frac{1}{p^{2} + {2p\quad{\sin\left( {\frac{{2k} - 1}{2n}\pi} \right)}} + 1}}},{n \in {odd}}} \\{{\prod\limits_{k = 1}^{n/2}\frac{1}{p^{2} + {2p\quad{\sin\left( {\frac{{2k} - 1}{2n}\pi} \right)}} + 1}},{n \in {even}}}\end{matrix},{p = {j\Omega}}} \right.} & {{Equation}\quad 4}\end{matrix}$

A second-order Butterworth transfer function to a fifth-orderButterworth transfer function are designed, and coefficients ofrespective terms are expressed in the following table when the transferfunction F(p) is represented by a rational function form as shown in thefollowing equation: $\begin{matrix}{{F(p)} = \frac{1}{\sum\limits_{k = 0}^{n}{d_{k}p^{k}}}} & {{Equation}\quad 5}\end{matrix}$ TABLE 1 coefficients of Butterworth transfer functionSecond Third Fourth Fifth d₀ 1.00000000000000 1.000000000000001.00000000000000 1.00000000000000 d₁ 1.41421356237310 2.000000000000002.61312592975275 3.23606797749979 d₂ =d₀ =d₁ 3.414213562373105.23606797749979 d₃ =d₀ =d₁ =d₂ d₄ =d₀ =d₁ d₅ =d₀ Power half 1 1 1 1frequency

A pole location of Butterworth filters is shown in FIG. 7, and afrequency response is indicated in FIG. 8. In FIG. 8, such a conditionthat a passband response has been enlarged 0.3 million times larger thanthe original passband response is additionally represented. It should benoted that details of the above-mentioned designing method of theButterworth filters are described in “STRUCTURE OF CIRCUIT NETWORK”written by Masamitsu Kawakami issued by KYORITSU Publisher in 1955.

As the second, an elliptic filter is listed up. In order to describe theelliptic filter, an elliptic function is required, so that this ellipticfunction will now be briefly described. First, the Jacobian ellipticfunction is defined by the following equation: $\begin{matrix}{{z = {\int_{0}^{x}\frac{\mathbb{d}x}{\sqrt{\left( {1 - x^{2}} \right)\left( {1 - {k^{2}x^{2}}} \right)}}}}{{{sn}\left( {z,k} \right)} = x}{{{cn}\left( {z,k} \right)} = \sqrt{1 - {{sn}\left( {z,k} \right)}^{2}}}{{{dn}\left( {z,k} \right)} = \sqrt{1 - {k^{2}{{sn}\left( {z,k} \right)}^{2}}}}} & {{Equation}\quad 6}\end{matrix}$

Subsequently, first kind complete elliptic integrals are defined by thefollowing equation: $\begin{matrix}{{K(k)} = {\int_{0}^{1}\frac{\mathbb{d}x}{\sqrt{\left( {1 - x^{2}} \right)\left( {1 - {k^{2}x^{2}}} \right)}}}} & {{Equation}\quad 7}\end{matrix}$

While various sorts of specific values of elliptic functions are known,the specific values required in this case are given as follows:$\begin{matrix}{{k_{2} = \sqrt{1 - k_{1}^{2}}},{{{sn}\left( {\frac{K\left( k_{2} \right)}{2},k_{2}} \right)} = \frac{1}{\sqrt{1 + k_{1}}}},{{{cn}\left( {\frac{K\left( k_{2} \right)}{2},k_{2}} \right)} = \sqrt{\frac{k_{1}}{1 + k_{1}}}}} & {{Equation}\quad 8}\end{matrix}$

While the above-mentioned items are prepared, various sorts ofparameters will now be defined which are required so as to describe thepole location of the elliptic filter. First of all, L₁ and L₂ aredefined from both a maximum passband loss “α” and a minimum stopbandloss “A” in accordance with the following equation: $\begin{matrix}{{L_{1} = \sqrt{\frac{\alpha^{2} - 1}{A^{2} - 1}}},{L_{2} = \sqrt{1 - L_{1}^{2}}}} & {{Equation}\quad 9}\end{matrix}$

Next, as solutions of the following equation, parameters “k₁” and “k₂”are defined. In this equation, “n” indicates an order of the transferfunction. It should be noted that the parameter “k₁” is called a“selectivity”, while there is such a relationship that a passband edgefrequency is √k₁, and a stopband edge frequency is 1/√k₁.$\begin{matrix}{{{k_{2} = \sqrt{1 - k_{1}^{2}}},n}{\frac{K\left( k_{2} \right)}{K\left( k_{1} \right)} = \frac{K\left( L_{2} \right)}{K\left( L_{1} \right)}}} & {{Equation}\quad 10}\end{matrix}$

Also, as a solution of the following equation, a parameter “ξ₀” isdetermined.sn(ξ₀ K(L ₂), L ₂)=1/α  Equation 11

The transfer function F(p) of the elliptic filter is given by thefollowing equation by employing the above-mentioned parameters:$\begin{matrix}{{a_{0} = \sqrt{k_{1}}}{\frac{{sn}\left( {{\xi_{0}{K\left( k_{2} \right)}},k_{2}} \right)}{{cn}\left( {{\xi_{0}{K\left( k_{2} \right)}},k_{2}} \right)},{\zeta_{k} = {\sqrt{k_{1}}{{sn}\left( {{\frac{n - {2k} + 1}{n}K_{1}},k_{1}} \right)}}}}{{\tau_{k} = \frac{{a_{0}\zeta_{k}} + \frac{1}{a_{0}\zeta_{k}}}{\sqrt{\left( {a_{0} - \frac{1}{a_{0}}} \right)^{2} + \left( {\zeta_{k} + \frac{1}{\zeta_{k}}} \right)^{2}}}},{Q_{k} = {\frac{1}{2}\sqrt{\frac{\left( {a_{0} - \frac{1}{a_{0}}} \right)^{2} + \left( {\zeta_{k} + \frac{1}{\zeta_{k}}} \right)^{2}}{\left( {\zeta_{k} + \frac{1}{\zeta_{k}}} \right)^{2} - \left( {\sqrt{k_{1}} + \frac{1}{\sqrt{k_{1}}}} \right)^{2}}}}}}} & {{Equation}\quad 12}\end{matrix}$

A detailed content as to the above-mentioned method of designing theelliptic filter is described in “APPROXIMATION AND STRUCTURE” written byMasamitsu Kawakami and Hiroshi Shibayama, published by KYORITSUpublisher in 1960.

In a general-purpose design of the elliptic filter, since the maximumpassband loss a and the minimum stopband loss A are independentlyselected, no pole is arranged on a unit circle. However, in the casewhere a restriction condition of Equation 13 is given between α and A, apole is arranged on the unit circle. FIG. 9 represents a graph in whicha relationship between α and A is plotted.α=A/√{square root over (A²−1)}  Equation 13

At this time, since the following equation can be established based uponEquation 11, it is determined that ξ₀=½ from the specific value of theelliptic function of Equation 8: $\begin{matrix}{{{sn}\left( {{\zeta_{0}{K\left( L_{2} \right)}},L_{2}} \right)} = {\frac{1}{\alpha} = \frac{1}{\sqrt{1 + L_{1}}}}} & {{Equation}\quad 14}\end{matrix}$

Furthermore, due to the specific value relationship of the ellipticfunction, α₀=1, and since α₀=1, T_(k)=1. $\begin{matrix}{\begin{matrix}{{a_{0} = \sqrt{k_{1}}}\frac{{sn}\left( {{{K\left( k_{2} \right)}/2},k_{2}} \right)}{{cn}\left( {{{K\left( k_{2} \right)}/2},k_{2}} \right)}} \\{= {\sqrt{k_{1}}{\frac{1}{\sqrt{1 + k_{1}}}/\sqrt{\frac{k_{1}}{1 + k_{1}}}}}} \\{= 1}\end{matrix}{\tau_{k} = 1}} & {{Equation}\quad 15}\end{matrix}$

As described above, also in the elliptic filter, since the restrictioncondition of Equation 13 is added, all of the poles are represented asbeing arranged on the unit circle of the complex plane.

In this case, with respect to 3 conditions in which the minimum stopbandloss “A” is 30 dB, 40 dB, and 50 dB, when a second-order elliptic filtertransfer function F(p) to a fifth-order elliptic filter transferfunction F(p) are designed which satisfy Equation 13, and are expressedby a rational function form as shown in the following equation,coefficients of the respective terms are indicated in the followingtables. It should be noted that odd-order terms of a numerator are notpresent, and are therefore omitted. Also, the passband edge frequency“√k₁” and the stopband edge frequency “1/√k₁” are additionallyexpressed: $\begin{matrix}{{F(p)} = \frac{\sum\limits_{k = 0}^{n}{n_{k}p^{k}}}{\sum\limits_{k = 0}^{n}{d_{k}p^{k}}}} & {{Equation}\quad 16}\end{matrix}$

Locations of poles of the above-mentioned elliptic filter are indicatedin FIGS. 10 to 12, and frequency responses thereof are represented inFIGS. 13 to 15. Also, such conditions that passband responses have beenenlarged 3,000, 30,000, and 0.3 million times larger than the originalpassband responses, respectively, are additionally represented in FIGS.13 to 15: TABLE 2 coefficients of elliptic filter transfer function (A =30 dB) Second Third Fourth Fifth n₀ 0.99949987493746 1.000000000000000.99949987493746 1.00000000000000 n₂ 0.03162277660168 0.187558606480780.48766999793704 0.88982067836666 n₄ 0.03162277660168 0.16879048294331d₀ 1.00000000000000 1.00000000000000 1.00000000000000 1.00000000000000d₁ 1.36878491917677 1.81244139351922 2.12871347620239 2.34416832262638d₂ =d₀ =d₁ 2.76855811323175 3.62313812720303 d₃ =d₀ =d₁ =d₂ d₄ =d₀ =d₁d₅ =d₀ Passband 0.251424 0.498116 0.687714 0.814772 edge Stopband3.97735 2.00757 1.45409 1.22734 edge

TABLE 3 coefficients of elliptic filter transfer function (A = 40 dB)Second Third Fourth Fifth n₀ 0.99994999874994 1.000000000000000.99994999874994 1.00000000000000 n₂ 0.01000000000000 0.087573633615650.28004915242507 0.58396659682018 n₄ 0.01000000000000 0.07010011281415d₀ 1.00000000000000 1.00000000000000 1.00000000000000 1.00000000000000d₁ 1.40000071430339 1.91242636638435 2.34495203998657 2.67608110309858d₂ =d₀ =d₁ 3.03223567606022 4.16221461909255 d₃ =d₀ =d₁ =d₂ d₄ =d₀ =d₁d₅ =d₀ Passband 0.141418 0.341417 0.526583 0.673852 edge Stopband7.07124 2.92897 1.89903 1.48401 edge

TABLE 4 coefficients of elliptic filter transfer function (A = 50 dB)Second Third Fourth Fifth n₀  0.99999499998750 1.000000000000000.99999499998750 1.00000000000000 n₂  0.00316227766017 0.040701403964630.15855315179443 0.37465098565403 n₄ 0.00316227766017 0.02839050382652d₀  1.00000000000000 1.00000000000000 1.00000000000000 1.00000000000000d₁  1.40973435533261 1.95929859603537 2.46389807479992 2.88652672542544d₂ =d₀ =d₁ 3.19445060961793 4.54026624359793 d₃ =d₀ =d₁ =d₂ d₄ =d₀ =d₁d₅ =d₀ Passband  0.0795269 0.232913 0.39756 0.54435 edge Stopband12.5744 4.29344 2.51534 1.83705 edge

Next, a complex transfer function “G(s)” is derived. In general, atransfer function “F(p)” of a prototype lowpass characteristic having apole on a unit circle is expressed by the following equation:$\begin{matrix}{{F(p)} = \left\{ \begin{matrix}{{\frac{1}{p + 1}{\prod\limits_{k = 1}^{{({n - 1})}/2}\frac{{\zeta_{k}^{2}p^{2}} + 1}{p^{2} + \frac{p}{Q_{k}} + 1}}},{n \in {odd}}} \\{{A_{0}{\prod\limits_{k = 1}^{n/2}\frac{{\zeta_{k}^{2}p^{2}} + 1}{p^{2} + \frac{p}{Q_{k}} + 1}}},{n \in {even}}}\end{matrix} \right.} & {{Equation}\quad 17}\end{matrix}$

A bilinear variable transformation is defined based upon Equation 18. IfEquation 18 is solved with respect to “ω”, then responses with respectto Ω=0 and Ω=∞ of a prototype lowpass characteristic “H(p)” are mappedto ω=1 and ω=−1, respectively. As a consequence, such a new transferfunction G(s) that an area in the vicinity of ω=1 corresponds to apassband and an area in the vicinity of ω=−1 corresponds to a stopbandis obtained. Similarly, responses with respect to power half valuefrequencies Ω=−1 and Ω=1 of the Butterworth filter are mapped to ω=0 andω=∞, respectively. A passband edge frequency Ω=±√k₁ and a stopband edgefrequency Ω=±1/√k₁ of the elliptic filter are mapped toω=(1±√k₁)/(1∓√k₁) and ω=(±1√k₁)/(□1+√k₁), respectively. $\begin{matrix}{{p = {{j\quad\Omega} = j}}{\frac{\omega - 1}{\omega + 1} = j}{\frac{s - j}{s + j},{\omega = \frac{1 + \Omega}{1 - \Omega}}}} & {{Equation}\quad 18}\end{matrix}$

Since the prototype lowpass transfer function F(p) can be decomposed toa product of a first-order transfer function and a second-order transferfunction, respective function cases thereof will now be described.First, in case of the first-order transfer function, this function canbe modified as a form having a negative real pole and a negative zero onan imaginary axis: $\begin{matrix}{{{F(p)} = \frac{1}{p + 1}}{{G(s)} = \frac{\left( {1 + j} \right)\left( {{{- j}\quad s} + 1} \right)}{2\left( {s + 1} \right)}}} & {{Equation}\quad 19}\end{matrix}$

Next, in case of the second-order transfer function, it is conceivablethat the second-order function has a complex pole and thus may bedefined as Q>½, the second-order function may be modified as a formhaving two different negative real poles and two zeros on an imaginaryaxis: $\begin{matrix}{{{{\text{~~~~}{F(p)}} = \frac{{\zeta^{2}p^{2}} + 1}{p^{2} + \frac{p}{Q} + 1}},{\zeta < 1},{Q > \frac{1}{2}}}{{G(s)} = \frac{j\quad{Q\left( {1 - \zeta^{2}} \right)}\left( {{{- j}\quad s} + \frac{1 + \zeta}{1 - \zeta}} \right)\left( {{{- j}\quad s} + \frac{1 - \zeta}{1 + \zeta}} \right)}{\left( {s + {2Q} - \sqrt{{4Q^{2}} - 1}} \right)\left( {s + {2Q} + \sqrt{{4Q^{2}} - 1}} \right)}}} & {{Equation}\quad 20}\end{matrix}$

It should be noted that an inverse number relationship of the followingequation can be established: $\begin{matrix}{{{\left( \frac{1 + \zeta}{1 - \zeta} \right)\left( \frac{1 - \zeta}{1 + \zeta} \right)} = 1}{{\left( {{2Q} - \sqrt{{4Q^{2}} - 1}} \right)\left( {{2Q} + \sqrt{{4Q^{2}} - 1}} \right)} = 1}} & {{Equation}\quad 21}\end{matrix}$

As apparent from the foregoing description, the higher-order functionmay be expressed by the following form: $\begin{matrix}{{G(s)} = \left\{ \begin{matrix}{{\frac{\left( {1 + j} \right)\left( {{{- j}\quad s} + 1} \right)}{2\left( {s + 1} \right)}{\prod\limits_{k = 1}^{{({n - 1})}/2}\frac{j\quad{Q_{k}\left( {1 - \zeta_{k}^{2}} \right)}\left( {{{- j}\quad s} + \frac{1 + \zeta_{k}}{1 - \zeta_{k}}} \right)\left( {{{- j}\quad s} + \frac{1 - \zeta_{k}}{1 + \zeta_{k}}} \right)}{\left( {s + {2Q_{k}} - \sqrt{{4Q_{k}^{2}} - 1}} \right)\left( {s + {2Q_{k}} + \sqrt{{4Q_{k}^{2}} - 1}} \right)}}},{n \in {odd}}} \\{{\frac{1}{\alpha}{\prod\limits_{k = 1}^{n/2}\frac{j\quad{Q_{k}\left( {1 - \zeta_{k}^{2}} \right)}\left( {{{- j}\quad s} + \frac{1 + \zeta_{k}}{1 - \zeta_{k}}} \right)\left( {{{- j}\quad s} + \frac{1 - \zeta_{k}}{1 + \zeta_{k}}} \right)}{\left( {s + {2Q_{k}} - \sqrt{{4Q_{k}^{2}} - 1}} \right)\left( {s + {2Q_{k}} + \sqrt{{4Q_{k}^{2}} - 1}} \right)}}},{n \in {even}}}\end{matrix} \right.} & {{Equation}\quad 22}\end{matrix}$

Next, a transfer function “G_(n)(s)” normalized in such a manner that aresponse in a DC becomes G(0)=1 is expressed by the following equation:$\begin{matrix}{{G_{n}(s)} = \left\{ \begin{matrix}{{\frac{{{- j}\quad s} + 1}{s + 1}{\prod\limits_{k = 1}^{{({n - 1})}/2}\frac{\left( {{{- j}\quad s} + \frac{1 + \zeta_{k}}{1 - \zeta_{k}}} \right)\left( {{{- j}\quad s} + \frac{1 - \zeta_{k}}{1 + \zeta_{k}}} \right)}{\left( {s + {2Q_{k}} - \sqrt{{4Q_{k}^{2}} - 1}} \right)\left( {s + {2Q_{k}\sqrt{{4Q_{k}^{2}} - 1}}} \right)}}},{n \in {odd}}} \\{{\prod\limits_{k = 1}^{n/2}\frac{\left( {{{- j}\quad s} + \frac{1 + \zeta_{k}}{1 - \zeta_{k}}} \right)\left( {{{- j}\quad s} + \frac{1 - \zeta_{k}}{1 + \zeta_{k}}} \right)}{\left( {s + {2Q_{k}} - \sqrt{{4Q_{k}^{2}} - 1}} \right)\left( {s + {2Q_{k}} + \sqrt{{4Q_{k}^{2}} - 1}} \right)}},{n \in {even}}}\end{matrix} \right.} & {{Equation}\quad 23}\end{matrix}$

Now, results of deriving the complex transfer functions G_(n)(s) fromthe prototype lowpass transfer function F(p) shown in Tables 1 to 4 arerepresented. Tables 5 to 8 represent “ρ_(k)” and “σ_(k)” in the order ofmagnitudes when transfer functions which constitute rational functionsare indicated in a factorization form as shown in the followingequation. Also, both passband edge frequencies and stopband edgefrequencies are additionally expressed. FIGS. 16 to 19 show frequencyresponses of the complex transfer functions G_(n)(s). The complextransfer functions G_(n)(s) derived in accordance with theabove-mentioned sequential manners have the same forms as a transferfunction “H(s)” of an RC polyphase filter described next, and areextremely suitable when those complex transfer functions G_(n)(s) arerealized by the RC polyphase filters: $\begin{matrix}{{G_{n}(s)} = \frac{{\prod\limits_{k = 1}^{n}{{- j}\quad s}} + \rho_{k}}{{\prod\limits_{k = 1}^{n}s} + \sigma_{k}}} & {{Equation}\quad 24}\end{matrix}$ TABLE 5 coefficients of complex transfer function G_(n)(s)based upon Butterworth filter transfer function Second Third FourthFifth ρ₁ 1.00000000000000 1.00000000000000 1.000000000000001.00000000000000 ρ₂ =1/ρ₁ 1.00000000000000 1.000000000000001.00000000000000 ρ₃ =1/ρ₁ =1/ρ₂ 1.00000000000000 ρ₄ =1/ρ₁ =1/ρ₂ ρ₅ =1/ρ₁σ₁ 0.41421356237310 0.26794919243112 0.19891236737966 0.15838444032454σ₂ =1/σ₁ 1.00000000000000 0.66817863791930 0.50952544949443 σ₃ =1/σ₁=1/σ₂ 1.00000000000000 σ₄ =1/σ₁ =1/σ₂ Power half 0 0 0 0 frequency ∞ ∞ ∞∞

TABLE 6 coefficients of complex transfer function G_(n)(s) based uponelliptic filter transfer function (A = 30 dB) Second Third Fourth Fifthρ₁   0.69797675554639   0.39559514072625   0.21874683001089  0.12067501274822 ρ₂ =1/ρ₁   1.00000000000000   0.56558329813716  0.31267570049614 ρ₃ =1/ρ₁ =1/ρ₂   1.00000000000000 ρ₄ =1/ρ₁ =1/ρ₂ ρ₅=1/ρ₁ σ₁   0.39580532641247   0.21226147136004   0.11676725879284  0.06438566038773 σ₂ =1/σ₁   1.00000000000000   0.53741012426121  0.29569844986088 σ₃ =1/σ₁ =1/σ₂   1.00000000000000 σ₄ =1/σ₁ =1/σ₂ σ₅=1/σ₁ Passband   1.67174   2.98498   5.40438   9.79753 edge   0.59818  0.33501   0.185035   0.102067 Stopband −1.67174 −2.98498 −5.40438−9.79753 edge −0.59818 −0.33501 −0.185035 −0.102067

TABLE 7 coefficients of complex transfer function G_(n)(s) based uponelliptic filter transfer function (A = 40 dB) Second Third Fourth Fifthρ₁   0.81817768570152   0.54329510726822   0.34416957673178  0.21634433303010 ρ₂ =1/ρ₁   1.00000000000000   0.65980337319471  0.41748312274676 ρ₃ =1/ρ₁ =1/ρ₂   1.00000000000000 P₄ =1/ρ₁ =1/ρ₂ P₅=1/ρ₁ σ₁   0.40836765910767   0.24139917515004   0.15012686492035  0.09409853793590 σ₂ =1/σ₁   1.00000000000000   0.59497034502798  0.37045133579411 σ₃ =1/σ₁ =1/σ₂   1.00000000000000 σ₄ =1/σ₁ =1/σ₁ σ₅=1/σ₁ Passband   1.32942   2.03682   3.22461   5.13218 edge   0.752207  0.49096   0.310115   0.194849 Stopband −1.32942 −2.03682 −3.22461−5.13218 edge −0.752207 −0.49096 −0.310115 −0.194849

TABLE 8 coefficients of complex transfer function G_(n)(s) based uponelliptic filter transfer function (A = 50 dB) Second Third Fourth Fifthρ₁   0.89351931756594   0.66424534282358   0.46237671890669  0.31680598422688 ρ₂ =1/ρ₁   1.00000000000000   0.73466322053635  0.50970592309918 ρ₃ =1/ρ₁ =1/ρ₂   1.00000000000000 ρ₄ =1/ρ₁ =1/ρ₂ ρ₅=1/ρ₁ σ₁   0.41236235079214   0.25547773338925   0.17068162020138  0.11590091661920 σ₂ =1/σ₁   1.00000000000000   0.62718707448377  0.42041290766453 σ₃ =1/σ₁ =1/σ₂   1.00000000000000 σ₄ =1/σ₁ =1/σ₂ σ₅=1/σ₁ Passband   1.1728   1.60727   2.31983   3.38934 edge   0.852663  0.622174   0.431065   0.295043 Stopband −1.1728 −1.60727 −2.31983−3.38934 edge −0.852663 −0.622174 −0.431065 −0.295043

FIG. 20 indicates an example of a fifth-order RC polyphase filter. Inthis example, 5 stages of the RC polyphase filter 108 shown in FIG. 5are connected in a cascade connection manner. It should be noted thatsuffixes “a” to “d” imply same element values, for instance,R_(1a)=R_(1b)=R_(1c)=R_(1d)=R₁. In the case of realizing a fourth-ordercharacteristic, it may be conceived that R₅ is shortcircuited, and C₅ isopened. This technical idea may be similarly applied to a third-ordercharacteristic and a second-order characteristic.

A chain matrix “m_(i)” of the RC polyphase filter per stage is given bythe following equation. When “n” stages of the RC polyphase filters areconnected in the cascade connection manner, the resultant RC polyphasefilters become a product “M” of the respective chain matrixes, and atransfer function H(s) is obtained by an inverse number of a 1-row1-column element of the product “M.” A denominator D(s) constitutes ann-order polynomial of “s.” However, there is such a restriction that thedenominator D(s) can have only a negative real simple root “−λA_(i)”,while the restriction is caused by a passive RC circuit. It should benoted that when it is considered that all capacitors are opened in a DCin view of a nature of a circuit, the DC gain becomes H(0)=1.$\begin{matrix}{{{m_{i} = {\frac{1}{{{- j}\quad s} + \omega_{i}}\begin{pmatrix}{s + \omega_{i}} & {R_{i}\omega_{i}} \\{2\frac{s}{R_{i}}} & {s + \omega_{i}}\end{pmatrix}}},{\omega_{i} = \frac{1}{C_{i}R_{i}}}}{M = {\prod\limits_{i = 1}^{n}m_{i}}}{{H(s)} = {\frac{1}{M_{11}} = {\frac{{\prod\limits_{i = 1}^{n}{{- j}\quad s}} + \omega_{i}}{D(s)} = {\prod\limits_{i = 1}^{n}\frac{{{- j}\quad s} + \omega_{i}}{s + \lambda_{i}}}}}}} & {{Equation}\quad 25}\end{matrix}$

In the beginning, a numerator polynomial is determined. “ω” may bedirectly adapted with respect to “ρ” which has already been designed. Asto a denominator polynomial D(s), “n” sorts of resistance values must bedetermined.

First, the transfer function H(s) is a non-dimensional amount, and animpedance level is indefinite, so any one of the resistance values isarbitrarily determined. Similar to the case of the numerator polynomial,if “λ” may be directly adapted to “σ”, then the denominator polynomialD(s) becomes preferable. However, normally, the denominator polynomialD(s) has a form which can be hardly factorized. As a consequence, thedenominator polynomial is once expanded so as to equally arrangecoefficients from a first-order term up to an (n-1)-order term, and(n-1) pieces of simultaneous equations related to the remaining (n-1)sorts of resistance values may be solved.

On the other hand, in the above-described methods, several differentsorts of solutions are obtained depending upon corresponding sequencesbetween “ρ” and “ω.” There are some possibilities that a negativeresistance value may be obtained depending upon the correspondingsequence. However, a circuit cannot be realized by employing thisnegative resistance value. As a result, “ρ” should be combined with “ω”by repeating trial and error until a positive resistance value isobtained. If all of element values are positive, any of solutions may beemployed. However, if an extent of element values is narrow, an RCpolyphase filter is easily manufactured.

While such a case of a third-order elliptic filter of a minimum stopbandloss (=30 dB) is exemplified, a description is made of a specificexample of the above-mentioned design sequence. First, “ρ” is read fromTable 6, and the read “ρ” is adapted to “ω” in the following manner:ω₁=ρ₃=1/ρ₁ω₂=ρ₂=1ω₃=ρ₁=0.39559514072625   Equation 26

Also, “σ” is read from Table 6 so as to form the following polynomials:σ₁=0.21226147136004σ₂=1σ₃=1/σ₁(s+σ ₁)(s+σ ₂)(s+σ ₃)=s ³+5.92343s ²+5.92343s+1   Equation 27

On the other hand, the denominator polynomial D(s) of the RC polyphasefilter transfer function is calculated from the chain matrix M so as toobtain the following equation with respect to unknown variables R₁, R₂,and R₃. $\begin{matrix}{{D(s)} = {s^{3} + {\left( {{3.92343 + 2}{\frac{R_{2}}{R_{3}} + {5.05567\frac{R_{1}}{R_{2}}} + {5.05567\frac{R_{1}}{R_{3}}}}} \right)s^{2}} + {\left( {{3.92343 + {5.05567\frac{R_{2}}{R_{3}}} + 2}{\frac{R_{1}}{R_{2}} + {5.05567\frac{R_{1}}{R_{3}}}}} \right)s} + 1}} & {{Equation}\quad 28}\end{matrix}$

For example, when R₂=1, if the above-mentioned two polynomials arecalculated by equally arranging coefficients, then the following elementvalues are obtained: $\begin{matrix}{{{R_{1} = 0.241626},{C_{1} = {\frac{1}{\omega_{1}R_{1}} = 1.63722}}}\text{}{{R_{2} = 1.00000},{C_{2} = {\frac{1}{\omega_{2}R_{2}} = 1.00000}}}{{R_{3} = 4.13863},{C_{3} = {\frac{1}{\omega_{3}R_{3}} = 0.610791}}}} & {{Equation}\quad 29}\end{matrix}$

As other examples than the above-mentioned example, even when “ω” and“ρ” are combined with each other in the following manner, the elementvalues are obtained. It should be noted that in this designing example,solutions obtained from combinations except the above-listedcombinations are improper: $\begin{matrix}{{{\rho_{1} = \omega_{1}},{\rho_{2} = \omega_{2}},{\rho_{3} = \omega_{3}}}{{R_{1} = 0.610791},{C_{1} = {\frac{1}{\omega_{1}R_{1}} = 4.13863}}}\text{}{{R_{2} = 1.00000},{C_{2} = {\frac{1}{\omega_{2}R_{2}} = 1.00000}}}{{R_{3} = 1.63722},{C_{3} = {\frac{1}{\omega_{3}R_{3}} = 0.241626}}}} & {{Equation}\quad 30}\end{matrix}$

Tables 9 to 12 represent results obtained by designing one set ofrespective element values of each of RC polyphase filters whichcorrespond to the complex transfer functions shown in Tables 5 to 8:TABLE 9 element values of RC polyphase filter based upon Butterworthfilter Element Second Third Fourth Fifth R₁ 0.643594253497840.41421356087532 0.26263785564678 0.16375128723073 R₂ 1.553773972600201.00000000000000 0.72041067071902 0.52337021220961 R₃ short2.41421357110275 1.38809715158984 1.00000000000000 R₄ short short3.80752423346349 1.91069338046219 R₅ short short short 6.10682222357714C₁ 1.55377397260020 2.41421357110275 3.80752423346349 6.10682222357714C₂ 0.64359425349784 1.00000000000000 1.38809715158984 1.91069338046219C₃ open 0.41421356087532 0.72041067071902 1.00000000000000 C₄ open open0.26263785564678 0.52337021220961 C₅ open open open 0.16375128723073

TABLE 10 element values of RC polyphase filter based upon ellipficfilter (A = 30 dB) Element Second Third Fourth Fifth R₁ 0.753043971059680.24162597031569 0.33366794524870 0.41260857823334 R₂ 1.327943703729291.00000000000000 0.35246374955489 0.66068347447247 R₃ short4.13862797402734 2.83717120203952 1.00000000000000 R₄ short short2.99699151278867 1.51358409683011 R₅ short short short 2.42360448316828C₁ 1.90256149974913 1.63722111983061 5.29893920302088 20.0837309064003C₂ 0.52560718806297 1.00000000000000 0.62062220626902 4.84074744033116C₃ open 0.61079104580789 1.61128620584120 1.00000000000000 C₄ open open0.18871701706445 0.20657966818686 C₅ open open open 0.04979154543847

TABLE 11 element values of RC polyphase filter based upon ellipticfilter (A = 40 dB) Second Third Fourth Fifth R₁ 0.706483227737010.29546774333741 0.31370046094607  0.34254991318114 R₂ 1.415461769988761.00000000000000 0.43780378816174  0.58616216517890 R₃ short3.38446420142063 2.28412824886425  1.00000000000000 R₄ short short3.18775432138081  1.70601253271744 R₅ short short short 2.91928259655167 C₁ 1.73001756983233 1.83876283908604 4.8313701545915913.4936864565018 C₂ 0.57802881163624 1.00000000000000 0.78612745219849 4.08642275522316 C₃ open 0.54384392524326 1.27205836305982 1.00000000000000 C₄ open open 0.20698062206011  0.24471281115539 C₅open open open  0.07410873249676

TABLE 12 element values of RC polyphase filter based upon ellipticfilter (A = 50 dB) Second Third Fourth Fifth R₁ 0.679340536036080.33264060203185 0.30081837896762  0.29901339074397 R₂ 1.472015796135101.00000000000000 0.50122337594968  0.48482839457192 R₃ short3.00624756536561 1.99511844016707  1.00000000000000 R₄ short short3.32426497154830  2.06258546569443 R₅ short short short 3.34433182912619 C₁ 1.64743589410182 1.99688594298967 4.5248828015701010.5564035896676 C₂ 0.60700389227904 1.00000000000000 0.92249631819411 4.04661859363080 C₃ open 0.50077972831179 1.08401516654029 1.00000000000000 C₄ open open 0.22100019908869  0.24711990439968 C₅open open open  0.09472923155181

While the various sorts of designing examples have been described above,all of those element values have been normalized. As a result, in anactual apparatus, if ratios of the element values of the elements whichconstitute a circuit are maintained, then the effect achieved by thisembodiment may be obtained. Also, a center of the passband and a centerof the stopband are 1 rad/s=0.16 Hz, and −1 rad/s=−0.16 Hz,respectively. In order to apply those centers to an actual circuit,impedance scaling must be performed. First, as to a resistance value,since a standard resistance value is approximately 1 kΩ in view of acircuit design, the resistance value may be uniformly multiplied by, forinstance, 1,000. A capacitance value may be determined in accordancewith a relationship between a desirable wave and an image. If thecapacitance value is uniformly divided by, for example, 1 billion, thecenters of the passband and the stopband are multiplied by 1 million, sothat those centers are moved to 1 Mrad/s=160 kHz, and −1 Mrad/s=−160kHz, respectively, and thus, those calculated values become elementvalues suitable for such a case where the desirable wave is apart fromthe image by 320 kHz.

Although Embodiment 1 has described the designing example based upon theRC polyphase filters, there are other passive RC complex filters capableof realizing the complex transfer function G_(n)(s). As an example,other modes of passive RC complex filters capable of realizing athird-order complex elliptic filter (A=30 dB) are represented in FIGS.21 and 23. Suffixes “x” and “y” imply the same element value. As to twooutput terminals, phases thereof are merely inverted, so that any one ofthose output terminals may be selected.

Frequency responses of FIGS. 21 and 23 are indicated in FIGS. 22 and 24,respectively. In those drawings, upper stages show amplitude responses,middle stages indicate enlarged diagrams thereof, and lower stagesrepresent phase responses. The same responses as those of FIG. 17 areobtained except that a gain reference is lowered by 3 dB, and an offsetof a phase is different.

Since the RC polyphase filter of FIG. 20, or the passive RC complexfilter of FIG. 21 or 23, which are designed according to this invention,are employed in the Hartley receiver of FIG. 5, the image interferencecan be suppressed over the broadband.

The passband of the passive RC complex filter according to thisinvention becomes flat in the light of the maximum flat model if theButterworth characteristic is selected as the prototype lowpasscharacteristic, and also becomes flat in the light of the equi-ripplemodel if the elliptic filter is selected. Also, the stopband of thepassive RC complex filter becomes flat in the light of the maximum flatmodel if the Butterworth characteristic is selected as the prototypelowpass characteristic, and also becomes flat in the light of theequi-ripple model if the simultaneous Chebyshev characteristic isselected. As a consequence, by employing the passive RC complex filteraccording to this invention, the superior reception characteristic canbe obtained.

INDUSTRIAL APPLICABILITY

This invention can be applied to a radio communication receivingapparatus.

1. A method of designing a passive RC complex filter, comprising thesteps of: designing a lowpass transfer function F(p) having poles on acircumference of a unit circle in a complex plane, which corresponds toa real coefficient rational function of second-order or higher relatedto a complex variable “p”; deriving a complex transfer function G(s)whose numerator is a complex coefficient and denominator is a realcoefficient, and which corresponds to the same-order rational functionas the lowpass transfer function F(p) and has poles different from eachother on a negative real axis and a zero point on an imaginary axis onthe complex plane, by performing a transformation of a variable ofp=j(s−j)/(s+j) related to a complex variable “s” with respect to thelowpass transfer function F(p); and calculating a complex transferfunction H(s) of the passive RC complex filter of the same order as thetransfer function F(p) to determine element values of the passive RCcomplex filter by equally arranging coefficients of the complex transferfunction G(s) and the complex transfer function H(s).
 2. The method ofdesigning a passive RC complex filter according to claim 1, wherein thetransfer function F(p) comprises a Butterworth filter.
 3. The method ofdesigning a passive RC complex filter according to claim 1, wherein: thetransfer function F(p) comprises a elliptic filter; and a maximumpassband loss “α” and a minimum stopband loss “A” satisfy a relationalexpression defined by α=A/√(A²−1).
 4. The method of designing a passiveRC complex filter according to claim 1, wherein the passive RC complexfilter comprises an RC polyphase filter having a number of stages sameas the order of the complex transfer function G(s).
 5. A passive RCcomplex filter, which is which is formed by the method comprising thesteps of: designing a lowpass transfer function F(p) having poles on acircumference of a unit circle in a complex plane, which corresponds toa real coefficient rational function of second-order or higher relatedto a complex variable “p”; deriving a complex transfer function G(s)whose numerator is a complex coefficient and denominator is a realcoefficient, and which corresponds to the same-order rational functionas the lowmass transfer function F(p) and has poles different from eachother on a negative real axis and a zero point on an imaginary axis onthe complex plane, by performing a transformation of a variable ofp=j(s−j)/(s+j) related to a complex variable “s” with respect to thelowpass transfer function F(p); and calculating a complex transferfunction H(s) of the passive RC complex filter of the same order as thetransfer function F(p) to determine element values of the passive RCcomplex filter by equally arranging coefficients of the complex transferfunction G(s) and the complex transfer function H(s). 6-22. (canceled)